3.158 \(\int x^2 \sqrt {a-a \cos (x)} \, dx\)

Optimal. Leaf size=56 \[ -2 x^2 \cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}+8 x \sqrt {a-a \cos (x)}+16 \cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]

[Out]

8*x*(a-a*cos(x))^(1/2)+16*cot(1/2*x)*(a-a*cos(x))^(1/2)-2*x^2*cot(1/2*x)*(a-a*cos(x))^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3319, 3296, 2638} \[ -2 x^2 \cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}+8 x \sqrt {a-a \cos (x)}+16 \cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a - a*Cos[x]],x]

[Out]

8*x*Sqrt[a - a*Cos[x]] + 16*Sqrt[a - a*Cos[x]]*Cot[x/2] - 2*x^2*Sqrt[a - a*Cos[x]]*Cot[x/2]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps

\begin {align*} \int x^2 \sqrt {a-a \cos (x)} \, dx &=\left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int x^2 \sin \left (\frac {x}{2}\right ) \, dx\\ &=-2 x^2 \sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )+\left (4 \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int x \cos \left (\frac {x}{2}\right ) \, dx\\ &=8 x \sqrt {a-a \cos (x)}-2 x^2 \sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )-\left (8 \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \sin \left (\frac {x}{2}\right ) \, dx\\ &=8 x \sqrt {a-a \cos (x)}+16 \sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )-2 x^2 \sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 30, normalized size = 0.54 \[ 8 \left (x-\frac {1}{4} \left (x^2-8\right ) \cot \left (\frac {x}{2}\right )\right ) \sqrt {a-a \cos (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a - a*Cos[x]],x]

[Out]

8*Sqrt[a - a*Cos[x]]*(x - ((-8 + x^2)*Cot[x/2])/4)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a-a*cos(x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [A]  time = 0.37, size = 51, normalized size = 0.91 \[ 2 \, \sqrt {2} {\left (4 \, x \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) - {\left (x^{2} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - 8 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \cos \left (\frac {1}{2} \, x\right ) - 8 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a-a*cos(x))^(1/2),x, algorithm="giac")

[Out]

2*sqrt(2)*(4*x*sgn(sin(1/2*x))*sin(1/2*x) - (x^2*sgn(sin(1/2*x)) - 8*sgn(sin(1/2*x)))*cos(1/2*x) - 8*sgn(sin(1
/2*x)))*sqrt(a)

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maple [C]  time = 0.06, size = 69, normalized size = 1.23 \[ -\frac {i \sqrt {2}\, \sqrt {-a \left ({\mathrm e}^{i x}-1\right )^{2} {\mathrm e}^{-i x}}\, \left (4 i x \,{\mathrm e}^{i x}+x^{2} {\mathrm e}^{i x}-4 i x +x^{2}-8 \,{\mathrm e}^{i x}-8\right )}{{\mathrm e}^{i x}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a-a*cos(x))^(1/2),x)

[Out]

-I*2^(1/2)*(-a*(exp(I*x)-1)^2*exp(-I*x))^(1/2)/(exp(I*x)-1)*(4*I*x*exp(I*x)+x^2*exp(I*x)-4*I*x+x^2-8*exp(I*x)-
8)

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maxima [B]  time = 1.38, size = 100, normalized size = 1.79 \[ {\left ({\left (4 \, \sqrt {2} x \cos \relax (x) + {\left (\sqrt {2} x^{2} - 8 \, \sqrt {2}\right )} \sin \relax (x) - 4 \, \sqrt {2} x\right )} \cos \left (\frac {1}{2} \, \pi + \frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right ) - {\left (\sqrt {2} x^{2} - 4 \, \sqrt {2} x \sin \relax (x) + {\left (\sqrt {2} x^{2} - 8 \, \sqrt {2}\right )} \cos \relax (x) - 8 \, \sqrt {2}\right )} \sin \left (\frac {1}{2} \, \pi + \frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x)\right )\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a-a*cos(x))^(1/2),x, algorithm="maxima")

[Out]

((4*sqrt(2)*x*cos(x) + (sqrt(2)*x^2 - 8*sqrt(2))*sin(x) - 4*sqrt(2)*x)*cos(1/2*pi + 1/2*arctan2(sin(x), cos(x)
)) - (sqrt(2)*x^2 - 4*sqrt(2)*x*sin(x) + (sqrt(2)*x^2 - 8*sqrt(2))*cos(x) - 8*sqrt(2))*sin(1/2*pi + 1/2*arctan
2(sin(x), cos(x))))*sqrt(a)

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mupad [B]  time = 0.34, size = 71, normalized size = 1.27 \[ \frac {2\,\sqrt {a}\,\sqrt {1-\cos \relax (x)}\,\left (8\,\cos \relax (x)-x^2\,\cos \relax (x)+4\,x\,\sin \relax (x)-x^2+8+x\,4{}\mathrm {i}+\sin \relax (x)\,8{}\mathrm {i}-x^2\,\sin \relax (x)\,1{}\mathrm {i}-x\,\cos \relax (x)\,4{}\mathrm {i}\right )}{\sin \relax (x)-\cos \relax (x)\,1{}\mathrm {i}+1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a - a*cos(x))^(1/2),x)

[Out]

(2*a^(1/2)*(1 - cos(x))^(1/2)*(x*4i + 8*cos(x) + sin(x)*8i - x^2*cos(x) - x^2*sin(x)*1i - x*cos(x)*4i + 4*x*si
n(x) - x^2 + 8))/(sin(x) - cos(x)*1i + 1i)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {- a \left (\cos {\relax (x )} - 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a-a*cos(x))**(1/2),x)

[Out]

Integral(x**2*sqrt(-a*(cos(x) - 1)), x)

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